Problem: Simplify the following expression and state the condition under which the simplification is valid: $y = \dfrac{z^2 + 8z}{z^2 + 13z + 40}$
First factor the expressions in the numerator and denominator. $ \dfrac{z^2 + 8z}{z^2 + 13z + 40} = \dfrac{(z)(z + 8)}{(z + 5)(z + 8)} $ Notice that the term $(z + 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(z + 8)$ gives: $y = \dfrac{z}{z + 5}$ Since we divided by $(z + 8)$, $z \neq -8$. $y = \dfrac{z}{z + 5}; \space z \neq -8$